Density-functional exchange correlation through coordinate scaling in adiabatic connection and correlation hole.

The exact exchange-correlation functional ${\mathit{E}}_{\mathrm{xc}}$[n] must be approximated in density-functional theory for the computation of electronic properties. By the coupling-constant integration (adiabatic-connection) formula we know that ${\mathit{E}}_{\mathrm{xc}}$[n]=${\mathcal{F}}_{0}^{1}$(${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n]-U[n])d\ensuremath{\alpha}, where ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n] is the electron-electron repulsion energy of ${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}^{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{\ensuremath{\alpha}}}$, which is that wave function that yields the density n and minimizes 〈T^+\ensuremath{\alpha}V${\mathrm{^}}_{\mathit{e}\mathit{e}}$〉. Here \ensuremath{\alpha} is the coupling constant. Consequently, knowledge of the behavior of ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n] as a function of \ensuremath{\alpha} ensures knowledge of ${\mathit{E}}_{\mathrm{xc}}$[n]. With this in mind and for the purpose of approximating ${\mathit{E}}_{\mathrm{xc}}$, it was previously established that (\ensuremath{\partial}${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$/\ensuremath{\partial}\ensuremath{\alpha})\ensuremath{\le}0. The present paper reveals that ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n]=\ensuremath{\alpha}${\mathit{V}}_{\mathit{e}\mathit{e}}^{1}$[${\mathit{n}}_{1/\mathrm{\ensuremath{\alpha}}}$], where ${\mathit{n}}_{\mathrm{\ensuremath{\beta}}}$(x,y,z)=${\mathrm{\ensuremath{\beta}}}^{3}$n(\ensuremath{\beta}x,\ensuremath{\beta}y,\ensuremath{\beta}z), and where \ensuremath{\beta} is a coordinate scale factor.